The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers. (If Zermelo–Fraenkel set theory (ZFC) is consistent, then neither the continuum hypothesis nor its negation can be proven from ZFC.)
Some authors, including P. Suppes and J. Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set, in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition, the following are all equivalent:
Category:Basic concepts in infinite set theory Category:Cardinal numbers Category:Ordinal numbers
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